Prime numbers serve as building blocks in the mathematics of whole numbers. Evenly divisible only by themselves and one, primes are a rich source of speculative ideas that mathematicians often find simple to state but difficult to prove. The Goldbach conjecture (every integer greater than 5 is the sum of three primes) is a prime example. Mathematicians and other researchers have turned to computers to test the conjecture against larger and larger even numbers. "There are strong grounds for believing that Goldbach's conjecture is true, and it feels like just a matter of time before someone figures out how to prove it," says Joe Buhler of the Mathematical Sciences Research Institute in Berkeley, Calif. "The real justification is algorithmic. In figuring out how to carry out the computations that far, one has to extend and polish algorithmic programming techniques, and the nature of the scientific advance in this case is much more in algorithmics than in number theory."